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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b5">P</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b6">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font><br/> for <font color="Olive" title="b7">R</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b6">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b7">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b7">R</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/> for <font color="Olive" title="b2">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font><br/> for <font color="Olive" title="b3">R</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">R</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font><br/> for <font color="Olive" title="b2">R</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">R</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">R</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">R</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> )</span><br/>















<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:69"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span><br/>

<a NAME="E4:69"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="borsuk_6.html#T46" target="_self" title="BORSUK_6:th.46">Th46</a></i>;<br/>
<a NAME="E5:69"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="borsuk_6.html#T46" target="_self" title="BORSUK_6:th.46">Th46</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font>;<br/>
<b>set </b><font color="Maroon" title="c10">f</font> =  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> ;<br/>
<b>set </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>;<br/>
<a NAME="E6:69"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<a NAME="E1:69_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<a NAME="E2:69_1"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E60">A7</font></i>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E4:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="borsuk_6.html#D6" target="_self" title="BORSUK_6:def.6">Def6</a>, <a class="ref" href="borsuk_6.html#T46" target="_self" title="BORSUK_6:th.46">Th46</a>, <a class="ref" href="borsuk_6.html#T57" target="_self" title="BORSUK_6:th.57">Th57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:69_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2"/>
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 ) or <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1"><b>suppose </b></a><a NAME="E1:69_1_2_1"/><i><font color="Green" title="E61">A8</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T9" target="_self" title="BORSUK_6:th.9">Th9</a></i>;<br/><a NAME="E3:69_1_2_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_1_2_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E4:69_1_2_1"/><b>then </b><i><font color="Green" title="E62">A9</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">z</font> = 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="#E3:69_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="#E1:69_1_2_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="borsuk_6.html#D9" target="_self" title="BORSUK_6:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E4:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E4:69_1_2_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E1:69_1_2_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E5:69"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="#E1:69_1_2_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E7:69_1_2_1"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2"><b>suppose </b></a><a NAME="E1:69_1_2_2"/><i><font color="Green" title="E61">A10</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><a NAME="E2:69_1_2_2"/><b>then </b><i><font color="Green" title="E62">A11</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E3:69_1_2_2"/><b>then </b><i><font color="Green" title="E63">A12</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:69_1_2_2"/><i><font color="Green" title="E64">A13</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_1_2_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E3:69_1_2_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">z</font> = 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E2:69_1_2_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="borsuk_6.html#T6" target="_self" title="BORSUK_6:th.6">Th6</a></i>;<br/><a NAME="E7:69_1_2_2"/><i><font color="Green" title="E65">A14</font></i>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="#E4:69_1_2_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">w</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E1:69_1_2_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="borsuk_6.html#T7" target="_self" title="BORSUK_6:th.7">Th7</a></i>;<br/><a NAME="E9:69_1_2_2"/>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_1_2_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E10:69_1_2_2"/><b>then </b><i><font color="Green" title="E66">A15</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="#E3:69_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="#E1:69_1_2_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="borsuk_6.html#D9" target="_self" title="BORSUK_6:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E4:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E2:69_1_2_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E7:69_1_2_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">w</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">w</font>
<b>by </b><i><a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E10:69_1_2_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E5:69"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="#E1:69_1_2_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E12:69_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_3"><b>suppose </b></a><a NAME="E1:69_1_2_3"/><i><font color="Green" title="E61">A16</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><a NAME="E2:69_1_2_3"/><b>then </b><i><font color="Green" title="E62">A17</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E1:69_1_2_3"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="borsuk_6.html#T7" target="_self" title="BORSUK_6:th.7">Th7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:69_1_2_3"/>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_1_2_3"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E5:69_1_2_3"/><b>then </b><i><font color="Green" title="E63">A18</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">w</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E1:69_1_2_3"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="borsuk_6.html#T7" target="_self" title="BORSUK_6:th.7">Th7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="#E3:69_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="#E1:69_1_2_3"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="borsuk_6.html#D9" target="_self" title="BORSUK_6:def.9">Def9</a></i>
<br/>.= 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E4:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E5:69_1_2_3"><i><font color="Green" title="E63">A18</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">w</font>
<b>by </b><i><a class="txt" href="#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E3:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E5:69_1_2_3"><i><font color="Green" title="E63">A18</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E5:69"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="#E2:69_1_2_3"><i><font color="Green" title="E62">A17</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E8:69_1_2_3"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:69"/>
 <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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